A fuzzy approach for prioritization of pharmacies to improve mask distribution process during COVID-19 pandemic—a pilot study for İstanbul

Abstract

While the whole world struggles with the COVID-19 pandemic, there are many different measures taken by countries. In this sense, the distribution of free masks to citizens between the ages of 20–65 in Turkey is one of the important measures taken against to spread of the pandemic. This distribution process is carried out through pharmacies and people can obtain their masks from any pharmacy in their area of residence. However, this situation may cause some pharmacies to be very busy, and thus social distance cannot be maintained and health and safety of the people may be threatened. In this paper, we aim to prioritize pharmacies so that only determined pharmacies in certain regions perform mask distribution process to prevent virus transmission. For this purpose, Esenler district is taken into consideration for a pilot study which is one of the risky regions in terms of virus spread in Istanbul, Turkey. Multi-criteria decision-making approach (MCDM) is used because of the necessity of handling many factors in decision-making process and the contradiction of evaluation factors in the prioritization of pharmacies. In order to best model the uncertainty in the decision process, the MCDM approach is applied in a fuzzy environment. In addition, spherical fuzzy AHP and VIKOR MCDM approaches are used as novel hybrid method in this paper. As a result of spherical fuzzy multi-criteria analysis, the pharmacies that need to provide free mask distribution in the Esenler region have been successfully identified.

Appendices

Appendix

1.1 Preliminaries for spherical fuzzy sets

Intuitionistic fuzzy sets were suggested by Atanassov in 1986 to address allocating non-membership degrees [71]. The fuzzy sets extensions where a single element can have a number of values for the membership degree are named Hesitant Fuzzy Sets developed by Torra [72]. With the development of intuitionistic type-2 fuzzy sets, Pythagorean fuzzy sets were introduced in 2013 by Yager [73]. For the SFSs, at the time that the squared sum of membership, non-membership and hesitancy parameters can be between 0 and 1, each of them can be separately determined between 0 and 1 to ensure that the sum of squares equals at most 1 [74, 75]. SFSs definition and spherical distance measurement, arithmetic operations, aggregation operators and defuzzification operations are given as in the following definitions [13, 14, 75, 76].

Definition 1

Let U is a universe of discourse. By a spherical fuzzy set (SFS) of U, we shall understand a set of the form as defined follows [74]:

$$\tilde{A}_{s} = \left\{ {\left\langle {(\mu_{{\tilde{A}_{S} }} (u),\upsilon_{{\tilde{A}_{S} }} (u),\pi_{{\tilde{A}_{S} }} (u))} \right\rangle :u \in U} \right\}$$

(1)

where $\mu_{{\tilde{A}_{s} }} {,}\upsilon_{{\tilde{A}_{s} }} ,\pi_{{\tilde{A}_{s} }} :U \to [0,1]\,[0,1]$ are the functions that quantify the degree of membership, non-membership, and hesitancy of each $u \in U$ with respect to $\tilde{A}_{s}$, respectively. They must satisfy the condition

$$\mu_{{\tilde{A}_{s} }}^{2} (u) + \upsilon_{{\tilde{A}_{s} }}^{2} (u) + \pi_{{\tilde{A}_{s} }}^{2} (u) \in [0,1]{\text{ for all }}u \in U.$$

(2)

In what follows, we shall denote a given SFS, by $\tilde{A}_{s} = (\mu_{{\tilde{A}_{s} }}^{2} ,\upsilon_{{\tilde{A}_{s} }}^{2} \pi_{{\tilde{A}_{s} }}^{2} )$, for short.

The numbers $\mu_{{\tilde{A}_{s} }}^{2} ,\upsilon_{{\tilde{A}_{s} }}^{2}$ and $\pi_{{\tilde{A}_{s} }}^{2}$ are the degree of membership, nonmembership and hesitancy of u to $\tilde{A}_{s}$ for each u, respectively.

Definition 2

The main arithmetic operations can be defined as follows [74]:

Addition;

$$\tilde{A}_{s} \oplus \tilde{B}_{s} = \left\{ {\left( {\mu_{{\tilde{A}_{s} }}^{2} + \mu_{{\tilde{B}_{s} }}^{2} - \mu_{{\tilde{A}_{s} }}^{2} \mu_{{\tilde{B}_{s} }}^{2} } \right)^{1/2} ,\upsilon_{{\tilde{A}_{s} }} \upsilon_{{\tilde{B}_{s} }} ,\left( {\left( {1 - \mu_{{\tilde{B}_{s} }}^{2} } \right)\pi_{{\tilde{A}_{s} }}^{2} + \left( {1 - \mu_{{\tilde{A}_{s} }}^{2} } \right)\pi_{{\tilde{B}_{s} }}^{2} - \pi_{{\tilde{A}_{s} }}^{2} \pi_{{\tilde{B}_{s} }}^{2} } \right)^{1/2} } \right\}$$

(3)

Multiplication;

$$\tilde{A}_{s} \otimes \tilde{B}_{s} = \left\{ {\mu_{{\tilde{A}_{s} }} \mu_{{\tilde{B}_{s} }} ,\left( {\upsilon_{{\tilde{A}_{s} }}^{2} + \upsilon_{{\tilde{B}_{s} }}^{2} - \upsilon_{{\tilde{A}_{s} }}^{2} \upsilon_{{\tilde{B}_{s} }}^{2} } \right)^{1/2} ,\left( {\left( {1 - \upsilon_{{\tilde{B}_{s} }}^{2} } \right)\pi_{{\tilde{A}_{s} }}^{2} + \left( {1 - \upsilon_{{\tilde{A}_{s} }}^{2} } \right)\pi_{{\tilde{B}_{s} }}^{2} - \pi_{{\tilde{A}_{s} }}^{2} \pi_{{\tilde{B}_{s} }}^{2} } \right)^{1/2} } \right\}$$

(4)

Multiplication by a scalar; λ > 0

$$\lambda .\tilde{A}_{s} = \left\{ {\left( {1 - \left( {1 - \mu_{{\tilde{A}_{s} }}^{2} } \right)^{\lambda } } \right)^{1/2} ,\upsilon_{{\tilde{A}_{s} }}^{\lambda } ,\left( {\left( {1 - \mu_{{\tilde{A}_{s} }}^{2} } \right)^{\lambda } - \left( {1 - \mu_{{\tilde{A}_{s} }}^{2} - \pi_{{\tilde{A}_{s} }}^{2} } \right)^{\lambda } } \right)^{1/2} } \right\}$$

(5)

λ. Power of $\tilde{A}_{s}$; λ > 0

$$\tilde{A}_{S}^{\lambda } = \left\{ {\mu_{{\tilde{A}_{s} }}^{\lambda } ,\left( {1 - \left( {1 - \upsilon_{{\tilde{A}_{s} }}^{2} } \right)^{\lambda } } \right)^{1/2} ,\left( {\left( {1 - \upsilon_{{\tilde{A}_{s} }}^{2} } \right)^{\lambda } - \left( {1 - \upsilon_{{\tilde{A}_{s} }}^{2} - \pi_{{\tilde{A}_{s} }}^{2} } \right)^{\lambda } } \right)^{1/2} } \right\}$$

(6)

Definition 3

Let $\tilde{A}_{s}$ and $\tilde{B}_{s}$ be two SFSs. The following identities hold for every $\lambda ,\lambda_{1} ,\lambda_{2} > 0$ [16].

$${\text{i}}{.}\quad \tilde{A}_{s} \oplus \tilde{B}_{s} = \tilde{B}_{s} \oplus \tilde{A}_{s}$$

(7)

$${\text{ii}}{.}\quad \tilde{A}_{s} \otimes \tilde{B}_{s} = \tilde{B}_{s} \otimes \tilde{A}_{s}$$

(8)

$${\text{iii}}{.}\quad \lambda (\tilde{A}_{s} \oplus \tilde{B}_{s} ) = \lambda \tilde{A}_{s} \oplus \lambda \tilde{B}_{s}$$

(9)

$${\text{iv}}{.}\quad \lambda_{1} \tilde{A}_{s} \oplus \lambda_{2} \tilde{A}_{s} = (\lambda_{1} + \lambda_{2} )\tilde{A}_{s}$$

(10)

$${\text{v}}{.}\quad (\tilde{A}_{s} \otimes \tilde{B}_{s} )^{\lambda } = \tilde{A}_{s}^{\lambda } \otimes \tilde{B}_{s}^{\lambda }$$

(11)

$${\text{vi}}{.}\quad \tilde{A}_{s}^{{\lambda_{1} }} \otimes \tilde{A}_{s}^{{\lambda_{2} }} = \tilde{A}_{s}^{{\lambda_{1} + \lambda_{2} }}$$

(12)

Definition 4

Let $w = (w_{1} ,...,w_{n} )$ be a weighted list of weights, i.e., $w_{i} \in [0,1]$ for all i = 1,…,n with $\sum\nolimits_{i = 1}^{n} {w_{i} = 1.}$ The Spherical Weighted Arithmetic Mean (SWAM) of the SFSs $\tilde{A}_{{s_{1} }} ,\tilde{A}_{{s_{2} }} ,...,\tilde{A}_{{s_{n} }}$ related to $w$ is described as [74]:

$$\begin{gathered} SWAM_{w} (\tilde{A}_{{s_{1} }} ,\tilde{A}_{{s_{2} }} ,...,\tilde{A}_{{s_{n} }} ) = w_{1} \tilde{A}_{{s_{1} }} + w_{2} \tilde{A}_{{s_{2} }} + ... + w_{n} \tilde{A}_{{s_{n} }} \hfill \\ { = }\left\{ {\left[ {1 - \prod\limits_{i = 1}^{n} {(1 - \mu_{{\tilde{A}_{{S_{i} }} }}^{2} )^{{w_{i} }} } } \right]^{1/2} ,\prod\limits_{i = 1}^{n} {\upsilon_{{\tilde{A}_{{S_{i} }} }}^{{w_{i} }} ,\left[ {\prod\limits_{i = 1}^{n} {(1 - \mu_{{\tilde{A}_{{S_{i} }} }}^{2} )^{{w_{i} }} } - \prod\limits_{i = 1}^{n} {(1 - \mu_{{\tilde{A}_{{S_{i} }} }}^{2} - \pi_{{\tilde{A}_{{S_{i} }} }}^{2} )^{{w_{i} }} } } \right]}^{1/2} } \right\} \hfill \\ \end{gathered}$$

(13)

Definition 5

Spherical Weighted Geometric Mean (SWGM) in reference to $w = (w_{1} ,...,w_{n} ); \, w_{i} \in [0,1];$$\sum\nolimits_{i = 1}^{n} {w_{i} = 1,}$ SWGM is described as [74];

$$\begin{gathered} SWGM_{w} (\tilde{A}_{{s_{1} }} ,\tilde{A}_{{s_{2} }} ,...,\tilde{A}_{{s_{n} }} ) = \tilde{A}_{{s_{1} }}^{{w_{1} }} + \tilde{A}_{{s_{1} }}^{{w_{2} }} + ... + \tilde{A}_{{s_{n} }}^{{w_{n} }} \hfill \\ { = }\left\{ {\prod\limits_{i = 1}^{n} {\mu_{{\tilde{A}_{{S_{i} }} }}^{{w_{i} }} ,} \left[ {1 - \prod\limits_{i = 1}^{n} {(1 - \upsilon_{{\tilde{A}_{{S_{i} }} }}^{2} )^{{w_{i} }} } } \right]^{1/2} ,\left[ {\prod\limits_{i = 1}^{n} {(1 - \upsilon_{{\tilde{A}_{{S_{i} }} }}^{2} )^{{w_{i} }} - } \prod\limits_{i = 1}^{n} {(1 - \upsilon_{{\tilde{A}_{{S_{i} }} }}^{2} - \pi_{{\tilde{A}_{{S_{i} }} }}^{2} )^{{w_{i} }} } } \right]^{1/2} } \right\} \hfill \\ \end{gathered}$$

(14)

Spherical fuzzy AHP

The steps of the SF-AHP method are summarized as follows [13]:

Step 1 The hierarchical structure is constructed. Criteria and alternatives are defined. Criteria are shown as $C_{j} (j = 1,2,...,n)$ and alternatives are shown as $X_{i} (i = 1,2,....,m)$ where $m,n \in {\mathbb{N}}$ [13]. The main criteria and sub-criteria considered within the scope of this study and the pharmacies in the determined region as alternatives are shown in figure that placed in real case analysis section.

Step 2 Decision-makers are asked to construct spherical fuzzy judgment matrices using the linguistic terms in Table 11 for pairwise comparisons of the criteria. Equation (15) and Eq. (16) are also used to calculate the score indices (SIs).

$$SI = \sqrt {\left| {100*\left[ {\left( {\mu_{{\tilde{A}_{s} }} - \pi_{{\tilde{A}_{s} }} } \right)^{2} - \left( {\upsilon_{{\tilde{A}_{s} }} - \pi_{{\tilde{A}_{s} }} } \right)^{2} } \right]} \right|}$$

(15)

and

$$\frac{1}{SI} = \frac{1}{{\sqrt {\left| {100*\left[ {\left( {\mu_{{\tilde{A}_{s} }} - \pi_{{\tilde{A}_{s} }} } \right)^{2} - \left( {\upsilon_{{\tilde{A}_{s} }} - \pi_{{\tilde{A}_{s} }} } \right)^{2} } \right]} \right|} }}$$

(16)

Step 3 The consistency of all pairwise comparison matrices has been checked. For this purpose, linguistic terms in the pairwise comparison matrices are converted to their corresponding score index in Table 11.

Table 11 Linguistic terms set for pairwise comparisons

The consistency ratio is calculated by using Eqs. (17–18). If the consistency ratio (CR) is smaller than 0.1, then the pairwise comparison matrices are consistent.

$$CI = \frac{{\lambda_{\max } - n}}{n - 1}$$

(17)

$$CR = \frac{CI}{{RI}}$$

(18)

where the n is the number of criteria (i = 1,…,n) that are compared and RI is the random index which varies randomly according to the criteria/alternatives number.

Step 4 Once all matrices are determined consistently, the evaluations for each decision-maker using the SWGM operator given in Definition 5 are aggregated [13]. In this step, w corresponds to the weight of the decision-maker.

Step 5 The spherical fuzzy local weights of criteria are determined by applying SWAM operator given in Definition 4 concerning each criterion. The method of weighted arithmetic mean is applied to calculate the spherical fuzzy weights. In this step, w corresponds to 1/n (n = number of criteria).

Step 6 The hierarchical layer sequencing to obtain global spherical fuzzy weights of sub-criteria is created. The spherical fuzzy weights at each level are aggregated and this calculation is performed from the bottom level (sub-criteria) to the top level (goal) [13]. The researcher can choose two different options at this stage. The first one is to defuzzify the weights of criteria by utilizing the score function (S) in Eq. (19). After that, normalize the weights by applying Eq. (20) and use spherical fuzzy multiplication given in Eq. (21).

$$S(\tilde{w}_{j}^{s} ) = \sqrt {\left| {100*\left[ {\left( {3\mu_{{\tilde{A}_{s} }} - \frac{{\pi_{{\tilde{A}_{s} }} }}{2}} \right)^{2} - \left( {\frac{{\upsilon_{{\tilde{A}_{s} }} }}{2} - \pi_{{\tilde{A}_{s} }} } \right)^{2} } \right]} \right|}$$

(19)

$$\overline{w}_{j}^{s} = \frac{{S(\tilde{w}_{j}^{s} )}}{{\sum\nolimits_{j = 1}^{n} {S(\tilde{w}_{j}^{s} )} }}$$

(20)

$$\tilde{A}_{{S_{ij} }} = \overline{w}_{j}^{s} .\tilde{A}_{{S_{i} }} = \left\langle {\left( {1 - (1 - \mu_{{\tilde{A}_{S} }}^{2} )^{{\overline{w}_{j}^{s} }} } \right)^{1/2} ,\upsilon_{{\tilde{A}_{S} }}^{{\overline{w}_{j}^{s} }} ,\left( {(1 - \mu_{{\tilde{A}_{S} }}^{2} )^{{\overline{w}_{j}^{s} }} - (1 - \mu_{{\tilde{A}_{S} }}^{2} - \pi_{{\tilde{A}_{S} }}^{2} )^{{\overline{w}_{j}^{s} }} } \right)^{1/2} } \right\rangle \, \forall i$$

(21)

where $(i = 1,2,....,m)$.

The value of the final spherical fuzzy AHP score $(\tilde{F})$, for each alternative A_i, is achieved by applying the spherical fuzzy arithmetic addition over each global preference weight as presented in Eq. (22).

$$\tilde{F} = \sum\limits_{j = 1}^{n} {\tilde{A}_{{S_{ij} }} = \tilde{A}_{{S_{i1} }} \oplus \tilde{A}_{{S_{i2} }} \oplus ... \oplus \tilde{A}_{{S_{in} }} } \, \forall i$$

where $(i = 1,2,....,m)$.i.e.,

$$\tilde{A}_{{S_{11} }} \oplus \tilde{A}_{{S_{12} }} = \left\langle {\left( {\mu_{{\tilde{A}_{{S_{11} }} }}^{2} + \mu_{{\tilde{A}_{{S_{12} }} }}^{2} - \mu_{{\tilde{A}_{{S_{11} }} }}^{2} \mu_{{\tilde{A}_{{S_{12} }} }}^{2} } \right)^{1/2} ,\upsilon_{{\tilde{A}_{{S_{11} }} }} \upsilon_{{\tilde{A}_{{S_{12} }} }} ,\left( {(1 - \mu_{{\tilde{A}_{{S_{12} }} }}^{2} )\pi_{{A_{{S_{11} }} }}^{2} + (1 - \mu_{{\tilde{A}_{{S_{11} }} }}^{2} )\pi_{{A_{{S_{12} }} }}^{2} - \pi_{{A_{{S_{11} }} }}^{2} \pi_{{A_{{S_{12} }} }}^{2} } \right)^{1/2} } \right\rangle$$

(22)

The other option is to keep on without defuzzification. At that, spherical fuzzy global preference weights are calculated by applying Eq. (23), which means multiplicating the main criteria weights and corresponding sub-criteria weights [13].

$$\prod\limits_{j = 1}^{n} {\tilde{A}_{{S_{ij} }} = } \tilde{A}_{{S_{i1} }} \otimes \tilde{A}_{{S_{i2} }} \otimes ... \otimes \tilde{A}_{{S_{in} }} \, \forall i$$

where $(i = 1,2,....,m)$.i.e.,

$$\tilde{A}_{{S_{11} }} \otimes \tilde{A}_{{S_{12} }} = \left\langle {\mu_{{\tilde{A}_{{S_{11} }} }} \mu_{{\tilde{A}_{{S_{12} }} }} ,\left( {\upsilon_{{\tilde{A}_{{S_{11} }} }}^{2} + \upsilon_{{\tilde{A}_{{S_{12} }} }}^{2} - \upsilon_{{\tilde{A}_{{S_{11} }} }}^{2} \upsilon_{{\tilde{A}_{{S_{12} }} }}^{2} } \right)^{1/2} ,\left( {(1 - \upsilon_{{\tilde{A}_{{S_{12} }} }}^{2} )\pi_{{\tilde{A}_{{S_{11} }} }}^{2} + (1 - \upsilon_{{\tilde{A}_{{S_{11} }} }}^{2} )\pi_{{\tilde{A}_{{S_{12} }} }}^{2} - \pi_{{\tilde{A}_{{S_{11} }} }}^{2} \pi_{{\tilde{A}_{{S_{12} }} }}^{2} } \right)^{1/2} } \right\rangle$$

(23)

In this paper, we adopted the full fuzzy approach in calculating the fuzzy spherical weights of local and global weights of the criteria to reflect the uncertainty better in the decision-making process.

Spherical fuzzy VIKOR

After determining the weights of main criteria and sub-criteria, the spherical fuzzy VIKOR method steps [76] are implemented to obtain the ranking of the alternatives.

Step 7 Decision-makers are asked to make criteria-alternative evaluations using the linguistic terms presented in Table 12.

Table 12 Linguistic Terms Set for Criteria-Alternative Evaluations

Step 8 The spherical fuzzy decision matrix based on the opinions of decision-makers is established provided through consensus. The evaluation values of alternative $X_{i} (i = 1,2,....,m)$ with respect to the criterion $C_{j} (j = 1,2,...,n)$ by $C_{j} (\tilde{X}_{i} ) = (\mu_{ij} ,\upsilon_{ij} ,\pi_{ij} )$ and $\tilde{D} = (C_{j} (\tilde{X}_{i} ))_{mxn}$ is a spherical fuzzy decision matrix. For an MCDM problem with SFS, a decision matrix $\tilde{D} = (C_{j} (\tilde{X}_{i} ))_{mxn}$ should be produced as in Eq. (24).

$$\tilde{D} = (C_{j} (\tilde{X}_{i} ))_{mxn} = \left( \begin{gathered} (\mu_{11} ,\upsilon_{11} ,\pi_{11} ) \, (\mu_{12} ,\upsilon_{12} ,\pi_{12} ) \, ... \, (\mu_{1n} ,\upsilon_{1n} ,\pi_{1n} ) \hfill \\ (\mu_{21} ,\upsilon_{21} ,\pi_{21} ) \, (\mu_{22} ,\upsilon_{22} ,\pi_{22} ) \, ... \, (\mu_{2n} ,\upsilon_{2n} ,\pi_{2n} ) \hfill \\ \, :{ : }...{ :} \hfill \\ \, :{ : }...{ :} \hfill \\ (\mu_{m1} ,\upsilon_{m1} ,\pi_{m1} ) \, (\mu_{m2} ,\upsilon_{m2} ,\pi_{m2} ) \, (\mu_{mn} ,\upsilon_{mn} ,\pi_{mn} ) \hfill \\ \end{gathered} \right)$$

(24)

where $(j = 1,2,...,n)$ and $(i = 1,2,....,m)$.

Step 9 The aggregated decision matrix is defuzzified by using the score value function by using Eq. (25).

$$Score(C_{j} (\tilde{X}_{i} )) = (2\mu_{ij} - \pi_{ij} )^{2} - (\upsilon_{ij} - \pi_{ij} )^{2}$$

(25)

where $(j = 1,2,...,n)$ and $(i = 1,2,....,m)$.

Step 10 The Spherical Fuzzy Positive Ideal Solution (SF-PIS) and the Spherical Fuzzy Negative Ideal Solution (SF-NIS) are defined derived the score values achieved in Step 9. Equation (26) is applied to obtain the maximum scores in the decision matrix for the SF-PIS. The respective SF numbers are calculated as in Eq. (27) based upon the crisp maximum scores.

$${X}^{*}=\left\{\left({C}_{j},\underset{i=1,\dots ,m}{max} \left\{\mathrm{Score}\left({C}_{j}\left({\widetilde{X}}_{i}\right)\right)\right\}\right):j=1,\dots ,n\right\}$$

(26)

$$\tilde{X}^{*} = \left\{ {\left\langle {C_{1} ,(\mu_{1}^{*} ,\upsilon_{1}^{*} ,\pi_{1}^{*} )} \right\rangle ,\left\langle {C_{2} ,(\mu_{2}^{*} ,\upsilon_{2}^{*} ,\pi_{2}^{*} )} \right\rangle ,...,\left\langle {C_{n} ,(\mu_{n}^{*} ,\upsilon_{n}^{*} ,\pi_{n}^{*} )} \right\rangle } \right\}$$

(27)

where $(j = 1,2,...,n)$ and $(i = 1,2,....,m)$.

Equation (28) is utilized to determine the minimum scores in the decision matrix for the SF-NIS. Based upon the crisp minimum scores, the respective SF numbers are defined as in Eq. (29).

$$X^{ - } = \left\{ {C_{j} ,\mathop {\mathop {\min }\limits_{i} < Score(C_{j} (\tilde{X}_{i} ) > |j = 1,2,...,n)}\limits_{{}} } \right\}$$

(28)

$$\tilde{X}^{ - } = \left\{ {\left\langle {C_{1} ,(\mu_{1}^{ - } ,\upsilon_{1}^{ - } ,\pi_{1}^{ - } )} \right\rangle ,\left\langle {C_{2} ,(\mu_{2}^{ - } ,\upsilon_{2}^{ - } ,\pi_{2}^{ - } )} \right\rangle ,...,\left\langle {C_{n} ,(\mu_{n}^{ - } ,\upsilon_{n}^{ - } ,\pi_{n}^{ - } )} \right\rangle } \right\}$$

(29)

where $(j = 1,2,...,n)$ and $(i = 1,2,....,m)$.

Step 11 Maximum group utility and a minimum individual regret of an opponent strategy values, which are shown as S_i and R_i, are calculated by the relations according to partial fuzzy and full fuzzy approaches as in Eq. (30) and Eq. (31), respectively. For partial fuzzy approach, spherical fuzzy weights which are calculated with SF-AHP method are defuzzified as in Eq. (19) and normalized with Eq. (20)

$$S_{i} = \sum\limits_{j = 1}^{n} {\overline{w}_{j}^{s} .D = \sum\limits_{j = 1}^{n} {\overline{w}_{j}^{s} .\frac{{D(\tilde{X}_{ij} ,\tilde{X}_{j}^{*} )}}{{D(\tilde{X}_{j}^{ - } ,\tilde{X}_{j}^{*} )}}} }$$

(30)

$$R_{i} = \mathop {\max }\limits_{j} (\overline{w}_{j}^{s} .D) = \mathop {\max }\limits_{j} \left( {\overline{w}_{j}^{s} .\frac{{D(\tilde{X}_{ij} ,\tilde{X}_{j}^{*} )}}{{D(\tilde{X}_{j}^{ - } ,\tilde{X}_{j}^{*} )}}} \right)$$

(31)

where $(j = 1,2,...,n)$ and $(i = 1,2,....,m)$.

On the other hand, spherical fuzzy weights calculated with SF-AHP method can be utilized with the full fuzzy approach as in Eq. (32). In this paper, we continued with the fuzzy approach as shown in the following equation.

$$D.\tilde{w}_{j}^{s} = \left\{ {\left( {1 - \left( {1 - \mu_{{\overline{w}_{j}^{s} }}^{2} } \right)^{D} } \right)^{1/2} ,\upsilon_{{\overline{w}_{j}^{s} }}^{\lambda } ,\left( {\left( {1 - \mu_{{\overline{w}_{j}^{s} }}^{2} } \right)^{D} - \left( {1 - \mu_{{\overline{w}_{j}^{s} }}^{2} - \pi_{{\overline{w}_{j}^{s} }}^{2} } \right)^{D} } \right)^{1/2} } \right\}$$

(32)

where $(j = 1,2,...,n)$.

There are three possible distance equations to calculate the distance in Eq. (32). Euclidean distance (Eq. (33) and Eq. (34)), Xu and Zhang’s distance (Eq. (35) and Eq. (36)) and spherical distance (Eq. (37) and Eq. (38)) can be applied [14, 77, 78].

$$D_{E} (\tilde{X}_{ij} ,\tilde{X}_{j}^{*} ) = \sqrt {\left( {\left( {\mu_{{\tilde{X}_{ij} }} - \mu_{{\tilde{X}_{j}^{*} }} } \right)^{2} + \left( {\upsilon_{{\tilde{X}_{ij} }} - \upsilon_{{\tilde{X}_{j}^{*} }} } \right)^{2} + \left( {\pi_{{\tilde{X}_{ij} }} - \pi_{{\tilde{X}_{j}^{*} }} } \right)^{2} } \right)}$$

(33)

$$D_{E} (\tilde{X}_{j}^{ - } ,\tilde{X}_{j}^{*} ) = \sqrt {\left( {\left( {\mu_{{\tilde{X}_{j}^{ - } }} - \mu_{{\tilde{X}_{j}^{*} }} } \right)^{2} + \left( {\upsilon_{{\tilde{X}_{j}^{ - } }} - \upsilon_{{\tilde{X}_{j}^{*} }} } \right)^{2} + \left( {\pi_{{\tilde{X}_{j}^{ - } }} - \pi_{{\tilde{X}_{j}^{*} }} } \right)^{2} } \right)}$$

(34)

$$D_{XZ} (\tilde{X}_{ij} ,\tilde{X}_{j}^{*} ) = \frac{1}{2}\left( {\left| {\mu_{{\tilde{X}_{ij} }}^{2} - \mu_{{\tilde{X}_{j}^{*} }}^{2} } \right| + \left| {\upsilon_{{\tilde{X}_{ij} }}^{2} - \upsilon_{{\tilde{X}_{j}^{*} }}^{2} } \right| + \left| {\pi_{{\tilde{X}_{ij} }}^{2} - \pi_{{\tilde{X}_{j}^{*} }}^{2} } \right|} \right)$$

(35)

$$D_{XZ} (\tilde{X}_{j}^{ - } ,\tilde{X}_{j}^{*} ) = \frac{1}{2}\left( {\left| {\mu_{{\tilde{X}_{j}^{ - } }}^{2} - \mu_{{\tilde{X}_{j}^{*} }}^{2} } \right| + \left| {\upsilon_{{\tilde{X}_{j}^{ - } }}^{2} - \upsilon_{{\tilde{X}_{j}^{*} }}^{2} } \right| + \left| {\pi_{{\tilde{X}_{j}^{ - } }}^{2} - \pi_{{\tilde{X}_{j}^{*} }}^{2} } \right|} \right)$$

(36)

$$D_{S} (\tilde{X}_{ij} ,\tilde{X}_{j}^{*} ) = \frac{2}{\pi }\left( {\arccos \left( {\mu_{{\tilde{X}_{ij} }} \mu_{{\tilde{X}_{j}^{*} }} + \upsilon_{{\tilde{X}_{ij} }} \upsilon_{{\tilde{X}_{j}^{*} }} + \pi_{{\tilde{X}_{ij} }} \pi_{{\tilde{X}_{j}^{*} }} } \right)} \right)$$

(37)

$$D_{S} (\tilde{X}_{j}^{ - } ,\tilde{X}_{j}^{*} ) = \frac{2}{\pi }\left( {\arccos \left( {\mu_{{\tilde{X}_{j}^{ - } }} \mu_{{\tilde{X}_{j}^{*} }} + \upsilon_{{\tilde{X}_{j}^{ - } }} \upsilon_{{\tilde{X}_{j}^{*} }} + \pi_{{\tilde{X}_{j}^{ - } }} \pi_{{\tilde{X}_{j}^{*} }} } \right)} \right)$$

(38)

where $(j = 1,2,...,n)$ and $(i = 1,2,....,m)$. S_i represents the separation measure of alternative i from the best value, and R_i represents the separation measure of alternative I from the worst value.

$Score(D.\tilde{w}_{j}^{s} )$ can be calculated by using Eq. (20) and then S_i and R_i values can be found as in Eq. (39) and Eq. (40), respectively.

$$S_{i} = \sum\limits_{j = 1}^{n} {Score(D.\tilde{w}_{j}^{s} )}$$

(39)

$$R_{i} = \mathop {\max }\limits_{j} \left( {Score(D.\tilde{w}_{j}^{s} )} \right)$$

(40)

where $(j = 1,2,...,n)$ and $(i = 1,2,....,m)$.

Step 12 $S^{*} ,S^{ - } ,R^{*} ,R^{ - }$ and $Q_{i}$ values are calculated based on Eq. (41) and Eq. (42), respectively.

$$\begin{gathered} S^{*} = \mathop {\min }\limits_{i} S_{i} \hfill \\ S^{ - } = \mathop {\max }\limits_{i} S_{i} \hfill \\ R^{*} = \mathop {\min }\limits_{i} R_{i} \hfill \\ R^{ - } = \mathop {\max }\limits_{i} R_{i} \hfill \\ \end{gathered}$$

(41)

where $(j = 1,2,...,n)$ and $(i = 1,2,....,m)$.

$$Q_{i} = \nu \frac{{(S_{i} - S^{*} )}}{{(S^{ - } - S^{*} )}} + (1 - \nu )\frac{{(R_{i} - R^{*} )}}{{(R^{ - } - R^{*} )}}$$

(42)

where $(i = 1,2,....,m)$.

The indices min S_i and min R_i are concerned with maximum majority rule and minimum individual regret of an opponent strategy. Additionally, v is presented as the maximum group utility weight strategy and is usually presumed to be 0.5.

Step 13 The best alternative with the minimum of Q_i is determined and ranking of the alternatives is obtained. A compromise solution, the alternative (a′) which is ranked the best by the measure Q_i (minimum) is proposed if the following two conditions are satisfied:

Condition 1:“Acceptable advantage”:

$$Q(a^{^{\prime\prime}} ) - Q(a^{^{\prime}} ) \ge D(Q)$$

(43)

$$D(Q) = \frac{1}{m - 1}$$

(44)

where a’’ the alternative with the second position in the ranking list by m is the number of alternatives.

Condition 2: “Acceptable stability in decision making”: Alternative a’ must also be the best ranked by S or/and R.